If the subindex \(\omega = 1\) , we simply write \((i^{P^{m}}(\gamma , m), \nu ^{P^{m}}(\gamma , m))\) , and omit the subindex 1 when there is no confusion. In the sequel, we use the notions \((i(\gamma ), \nu (\gamma ))\)  and \((i(\gamma , m), \nu (\gamma , m))\)  to denote the Maslov-type index and the iterated index of symplectic path \(\gamma \)  with the periodic boundary condition which were introduced by Y. Long and his collaborators (cf.[1]}, [2]}, [3]}, [4]}, etc.).
